How to map an equirectangular image onto an icosphere in Blender 2.79b
I don't know how you could correctly unwrap an icosphere but you could do it another way around with an UV sphere, an icosphere, and a Shrinkwrap modifier:


Create an UV sphere.
Select all except the two poles and Unwrap > Sphere Projection mode, with Align to Object option.
Select the 2 poles and make a simple unwrap.
Prepare your material.
It works, your drawing is projected on your UV sphere.
In Object mode, create an icosphere within your UV sphere.
Give your UV sphere a Shrinkwrap modifier and choose the icosphere as Target.
Give your UV sphere one or probably 2 Subsurf modifiers, one above, one below the Shrinkwrap modifier. Test the best solution, and the best Subdivision values, it looks like an amount of 3 is necessary on the top Subsurf.
Now, thanks to the Shrinkwrap, your UV sphere has taken the shape of the icosphere, you should have the result you want.